This is a three part question and I am struggling to gets the correct answers.

Question

Part a: Assume that the height of your cylinder is 6 inches. Consider A as a function or r, so we can write that as A(r)=2π*r²+12π*r. What is the domain of A(r)? In other words, for which values of r is A(r) defined?

Part b: Write the r as the function of A. This is the inverse functions to A(r), i.e to turn A as a function of r into, as a function of A.

Part c: If the surface area is 150 square inches, then what is the radius r? In other words, evaluate r(150). Round your answer 2 decimal places.

Answer 1

Part a)

Can "r" by a negative number? No, although negative numbers are defined in math, real cylinders can't have a negative radius. Can "r" be zero? No, if it were, there would be no cylinder. It also makes sense that the upper limit would be restricted as well since, on earth, you can't have an infinitely large radius but it's really hard to define what's too big so I'd probably just say the domain is r > 0 (this can be written in many formats).

Part b)

A = 2πr²+12πr

A/(2π) = r² + 6r

A/(2π) + 9 = r² + 6r + 9

A/(2π) + 9 = (r + 3)²

√(A/(2π) + 9) = r + 3

r = √(A/(2π) + 9) - 3

r(A) = √(A/(2π) + 9) - 3

(Notice I only used the positive square root since we are being asked to find a function and +/- would not yield a function. Besides, we already stated that r > 0.)

Part c)

In part b we found r(A) = √(A/(2π) + 9) - 3

so r(150) = √(150/(2π) + 9) - 3 = √32.873 - 3 = 5.73 - 3 = 2.73 inches

Answer 2

A(r) must be a non-negative number. A negative number wouldn't make sense.

The surface area formula S(r)=2πr²+2πr(6) can be rewritten as S(r)=2πr(r + 6). Solving for r gives

S(r)/2π = r²+6r. Complete the square

S(r)/2π + 9 = r²+6r+9.

S(r)/2π + 9 =(r+3)². Square root (The negative can be ignored. This is a real world situation (see part a)

√(S(r)/2π + 9 ) = r + 3, so...

√(S(r)/2π + 9 ) - 3 = r.

√(150/2π + 9) - 3 = r

√(23.873 + 9) - 3 = r

√(32.873) - 3 = r

5.733 - 3 = r

2.733 = r

2.73 in ■